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1) Show that quadrilateral ABCD can be inscribed in a circle iff and are supplementary.

2) Prove that a parallelogram having perpendicular diagonals is a rhombus.

3) Prove that a parallelogram with equal diagonals is a rectangle.

4) Show that the diagonals of an isosceles trapezoid are equal.

5) A straight line cuts two concentric circles in points A, B, C and D in that order. AE and BF are parallel chords, one in each circle. If CG is perpendicular to BF and DH is perpendicular to AE, prove that .

6) Construct triangle ABC, given angle A, side AC and the radius r of the inscribed circle. Justify your construction.

7) Let a triangle ABC be right angled at C. The internal bisectors of angle A and angle B meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.

8) Three circles with radii , with . They are placed such that lies to the right of and touches it externally; lies to the right of and touches it externally. Further, there exist two straight lines each of which is a direct common tangent simultaneously to all the three circles. Find in terms of and .

17) Another corollary to above problem 15: The angle between two secants drawn to a circle from an exterior point is equal in degrees to half the difference of the two subtended arcs. Prove this.

18) A third corollary to above problem 15: The angle between two chords that intersect in the interior of a circle is equal in degrees to half the sum of the two subtended arcs. Prove this.

19) Theorem (Pythagoras): If a right triangle has arms of lengths a and b and its hypotenuse has length c, then . Prove this.

20) Corollary to above theorem: Given a triangle ABC, the angle at vertex C is a right angle iff side AB is a diameter of the circumcircle. Prove this.

21) Theorem: The angle between a chord and the tangent at one of its endpoints is equal in degrees to half the subtended arc. Prove.

22) Corollary to problem 21: The angle between a secant and a tangent meeting at a point outside a circle is equal in degrees to half the difference of the subtended arcs.

23) Fix an integer, . Given a circle, how should n points on this circle be chosen so as to maximize the area of the corresponding n-gon?

24) Theorem: Given and , suppose that and . Then, prove that and so . Prove this theorem.

25) Theorem: If , then the lengths of the corresponding sides of these two triangles are proportional. Prove.

26) The following lemma is important to prove the above theorem: Let U and V be points on sides AB and AC of . Then, UV is parallel to BC if and only if . You will have to prove this lemma as a part of the above proof.

27) Special case of above lemma: Let U and V be the midpoints of sides AB and AC, respectively in . Then, UV is parallel to BC and .

28) Suppose that the sides of are proportional to the corresponding sides of . Then, .

29) Given and , assume that and that . Then, .

30) Consider a non-trivial plane geometry question now: Let P be a point outside of parallelogram ABCD and . Prove that .

31) Given a circle and a point P not on the circle, choose an arbitrary line through P, meeting the circle at points X and Y. Then, the quantity depends only on the point P and is independent of the choice of the line through P.

32) You can given an alternative proof of Pythagoras’s theorem based on the following lemma: Suppose is a right triangle with hypotenuse AB and let CP be the altitude drawn to the hypotenuse. Then, . Prove both the lemma and based on it produce an alternative proof of Pythagorean theorem.

33) Prove the following: The three perpendicular bisectors of the sides of a triangle are concurrent at the circumcenter of the triangle.

34) Prove the law of sines.

35) Let R and K denote the circumradius and area of , respectively and let a, b and c denote the side lengths, as usual. Then, .

36) Theorem: The three medians of an arbitrary triangle are concurrent at a point that lies two thirds of the way along each median from the vertex of the triangle toward the midpoint of the opposite side.

37) Time to ponder: Prove: Suppose that in , medians BY and CZ have equal lengths. Prove that .

38) If the circumcenter and the centroid of a triangle coincide, then the triangle must be equilateral. Prove this fact.

39) Assume that is not equilateral and let G and O be its centroid and circumcentre respectively. Let H be the point on the Euler line GO that lies on the opposite side of G from O and such that . Then, prove that all the three altitudes of pass through H.

40) Prove the following basic fact about pedal triangles: The pedal triangles of each of the four triangles determined by an orthic quadruple are all the same.

41) Prove the following theorem: Given any triangle, all of the following points lie on a common circle: the three feet of the altitudes, the three midpoints of the sides, and the three Euler points. Furthermore, each of the line segments joining an Euler point to the midpoint of the opposite side is a diameter of this circle.

42) Prove the following theorem and its corollary: Let R be the circumradius of triangle ABC. Then, the distance from each Euler point of to the midpoint of the opposite side is R, and the radius of the nine-point circle of is . The corollary says: Suppose is not a right angled triangle and let H be its orthocentre. Then, , , , and have equal circumradii.

43) Prove the law of cosines.

44) Prove Heron’s formula.

45) Express the circumradius R of in terms of the lengths of the sides.

46) Prove that the three angle bisectors of a triangle are concurrent at a point I, equidistant from the sides of the triangle. If we denote the by r the distance from I to each of the sides, then the circle of radius r centered at I is the unique circle inscribed in the given triangle. Note that in order to prove this, the following elementary lemma is required to be proved: The bisector of angle ABC is the locus of points P in the interior of the angle that are equidistant from the sides of the triangle.

47) Given a triangle with area K, semiperimeter s, and inradius r, prove that . Use this to express r in terms of the lengths of the sides of the triangle.

Please be aware that the above set of questions is almost like almost like a necessary set of pre-requisites for RMO geometry. You have to master the basics first.